Nonlinear Dyn (2017) 88:1493–1500DOI 10.1007/s11071-016-3325-9

ORIGINAL PAPER

Can Lyapunov exponent predict critical transitionsin biological systems?

Fahimeh Nazarimehr · Sajad Jafari ·Seyed Mohammad Reza Hashemi Golpayegani ·J. C. Sprott

Received: 15 July 2016 / Accepted: 27 December 2016 / Published online: 7 January 2017© Springer Science+Business Media Dordrecht 2017

Abstract Transitions from one dynamical regime toanother one are observed in many complex systems,especially biological ones. It is possible that even aslight perturbation can cause such a transition. It isclear that this can happen to an object when it is closeto a tipping point. There is a lot of interest in find-ing ways to recognize that a tipping point (in which abifurcation occurs) is near. There is a possibility thatin complex systems, a phenomenon known as “criti-cal slowing down” may be used to detect the vicinityof a tipping point. In this paper, we propose Lyapunovexponents as an indicator of “critical slowing down.”

Keywords Critical transition · Early-warning signal ·Lyapunov exponent · Chaos

1 Introduction

Transitions from one dynamical regime to another oneare observed in complex systems such as dynamicaldisease, brain response to flickering light, climate, andfinancial markets [1–3]. Such “regime shifts” can be

F. Nazarimehr · S. Jafari (B) ·S. M. R. Hashemi GolpayeganiBiomedical Engineering Department, Amirkabir Universityof Technology, Tehran 15875-4413, Irane-mail: sajadjafari@aut.ac.ir

J. C. SprottDepartment of Physics, University of Wisconsin - Madison,Madison, WI 53706, USA

the result of a massive external shock, or a stepwisechange in the conditions. However, it is also possiblethat a slight perturbation can invoke a massive shift to acontrasting and lasting state. It is clear that this can hap-pen to an object when it is close to a tipping point. Astipping points can have large consequences and makemajor changes to the system’s behavior, there is muchinterest in finding ways to recognize that a bifurcationis near. There is a possibility that in complex systems,a phenomenon known as “critical slowing down” canbe used to detect the vicinity of a tipping point [1–4].Close to the tipping point, the return to equilibriumfrom small perturbations will become slower becausethe basin of attraction becomes shallower. Recent stud-ies suggest that critical slowing down typically causesan increase in the variance and temporal autocorrela-tion of fluctuations in the system states [4]. Figure1shows the variation of variance and autocorrelation atlag-1.

Chaos is a common feature in complex dynami-cal systems. Many systems from fields such as biol-ogy and economics exhibit chaos, and the study ofsuch systems and their signals has progressed in recentdecades. There has been an increasing interest in ana-lyzing neurophysiology from a nonlinear and chaoticsystems viewpoint in recent years [6–16]. It has beenclaimed that many biological systems, including thebrain (both in microscopic and macroscopic aspects)[17–23] and the heart [24,25], have chaotic properties.This is true as well for the atmosphere [26], voice sig-

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Fig. 1 i Bifurcation diagram in a bistable system. ii SD (blueplot) and autocorrelation at lag-1 (green plot) [5]. (Color figureonline)

nals [27], and electronic circuits [28,29]. For a dynami-cal system, sensitivity to initial conditions is quantifiedby Lyapunov exponents. Lyapunov exponents show theintrinsic instability of trajectories in a system and arecomputed as the average rate of exponential conver-gence or divergence of trajectories that are nearby inthe phase space. In two trajectories with nearby initialconditions on an attracting manifold, when the attrac-tor is chaotic, the trajectories diverge, on average, at an

exponential rate characterized by the largest Lyapunovexponent.Adiscrete-time systemwith all negativeLya-punov exponents will have an attracting fixed point orperiodic cycle and will not present chaotic behavior[30–33]. Lyapunov exponents can determine the flex-ibility of attractors in response to external perturba-tions. In other words, calculation of Lyapunov expo-nents locally along the attractor shows where a systemignores an external signal and where it responds to it[34–39]. For illustrating the mathematical definition ofLyapunov exponent, consider two points in a space, X0

and X0 +�X0. It is useful to study the mean exponen-tial rate of divergence of two initially close orbits usingthe formula,

λ = limt→∞

|�X0→0|

1

tln

|�X (X0, t)||�X0| (1)

This number is the Lyapunov exponent “λ” and is usedfor distinguishing between the various types of orbits[40]. To illustrate the properties of nonlinear dynamicalsystems, the Lyapunov exponent has a crucial role [32].Many approaches have been proposed to compute thismeasure based on system equations and time series [33,40–42]. In this paper, we suggest the largest Lyapunovexponent as an indicator of tipping points. We showhow sensitivity to initial conditions and recovery ratefrom small perturbations are connected.

2 Lyapunov exponent as a prominentearly-warning signal

In this paper, Lyapunov exponent is proposed as aprominent early-warning signal for predicting critical

Fig. 2 Bifurcation diagramof the salamander ERG bychanging flash frequency atcontrast c = 1 [49]

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Can Lyapunov exponent predict critical transitions in biological systems? 1495

Fig. 3 a Bifurcationdiagram and b Lyapunovexponent as a prominentearly-warning signal forpredicting criticaltransitions in brain responseto flickering light(τ = .058, B = 35,C = 1)

5 10 15 20 25 30 35 40-8

-7

-6

-5

-4

-3

-2

-1

0

1

Flash Frequency (Hz)

Lyap

unov

Exp

onen

t

a

b

transitions. However, there are some major challengesthat make this indicator hard to use. Computing Lya-punov exponents based on observed signals is difficult.It requires a long time series that is as clean as pos-sible (noise is a serious problem for calculating Lya-punov exponents) [25,43,44]. The first step in calcu-lating Lyapunov exponent is reconstruction of phasespace from a time series. Then, Lyapunov exponentscan be calculated using this reconstructed phase space[45–48]. In the rest of this section, the idea of antic-ipating tipping points is illustrated by two biologicalmodels and one ecological model.

2.1 Brain response to flickering light

Thefirst example is themodel of brain response toflick-ering light described in [49]. Periodic flashes of light

can be used to investigate dynamical properties of thevisual system. Crevier and Meister in 1998 proposed asimple iterative model that can show the same bifurca-tion as was observed in flicker vision of a salamander[49]. Figure2 shows a bifurcation diagram of the sala-mander ERG with respect to changing flash frequency.

This model, proposed in [49], is given by

yk+1 = e− 1f τ

(BC

1 + y4k+ yk

)

xk+1 = C

1 + y4k+1

(2)

where f is the flash frequency, τ is the time constant ofexponential decay, B is an amplification constant, C isthe stimulus contrast, y is the feedback variable, and xis the amplitude of response to a flash. The model usesnonlinear feedback to account for period doubling in

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Fig. 4 a Bifurcationdiagram and b Lyapunovexponent as a predictor oftipping points in ADDattention model with respectto changing A parameter(B = 5.821, w1 =1.487, w2 = 0.2223)

5 10 15 20 25 30-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

A

Lyap

unov

Exp

onen

t

a

b

the ERG response to periodic flashes. For more detailabout the model, see [49]. Figure3 shows when theLyapunov exponent reaches zero, there is a critical tran-sition. Red lines identify some tipping points and theirLyapunov exponents. Thus, we propose the Lyapunovexponent as a prominent early-warning signal for pre-dicting critical transitions.

2.2 Attention deficit disorder

The second example is an attention deficit disorder(ADD) model proposed in [50]. Dopamine deficiencyis one of the causes of this disorder. The model usesa simple nonlinear neuronal network which representsthe interactions of inhibitory and excitatory parts ofbrain action and is given by

xk+1 = B × tanh(w1xk) − A × tanh(w2xk) (3)

The Lyapunov exponent of this model demonstratesthat this measure can be a good predictor for tippingpoints (Fig. 4).

2.3 Consumption of resources

The third example is a continuous model of logisticgrowth and consumption of resources with a controlvariable grazing rate which is familiar from investiga-tion of critical slowing down dynamics [5,51,52]. Inthis model, resource biomass x grows logistically andis harvested according to

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Can Lyapunov exponent predict critical transitions in biological systems? 1497

a

b

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

c

Lyap

unov

Exp

onen

t

Fig. 5 a Bifurcation diagram of ecological model of growingresource under harvesting. b Lyapunov exponent of the model

dx =(r x

(1 − x

k

)− cx2

x2 + h2

)dt (4)

where r is the growth rate, K is the population’s carry-ing capacity, h is the half-saturation constant, and c isthe grazing rate. When c reaches the value c = 2.904,the ecosystem goes to an alternate state (critical transi-tion) through a fold bifurcation. Part a of Fig. 5 depicts abifurcation diagram of the model. Application of Lya-punov exponent as an early warning of critical tran-sitions in the continuous model of growing resourceunder harvesting is investigated in part b of Fig. 5. TheLyapunov exponent goes close to zero as the bifurca-tion approaches the tipping point.

In dynamical systems, sensitivity to initial condi-tions is an important feature that shows complexity ofa system. Lyapunov exponents measure sensitivity toinitial conditions. If a small perturbation is assumed asa change in initial conditions of a system, Lyapunovexponents determine the rate of divergence or conver-gence. The results show that the Lyapunov exponent isa prominent predictor of critical transitions.

3 Discussion

In order to show the ability of Lyapunov exponentin predicting tipping points, we compare it with theentropymethod. Entropy is ameasure of unpredictabil-ity [53]. In changing from one fixed point to anotherone, irregularity of the state of the system increasesbecause of critical slowing down. Thus, the entropymeasure shows a meaningful change. However, inmany other bifurcation points like changing fromperiod two to period four, the entropy does not showany monotonic change before the tipping point. Fig-ure6 shows the entropy of ADDmodel. Comparison ofFigs. 4 and 6 describes that the entropy measure onlyfinds transitions from one fixed point to another one,but the Lyapunov exponent depicts any transitions. Onthe other hand, Lyapunov exponent approaches zerobefore the parameter reaches the critical values. Thisproperty makes Lyapunov exponent a very good indi-cator of tipping points.

It is believed that Kolmogorov–Sinai entropy is thesum of the positive exponents Lyapunov exponents,which for a low-dimensional chaotic system is just thelargest LE since the others are zero or negative [44,54].This measure can be considered in future works.

4 Conclusion

In this paper, we proposed Lyapunov exponents asprominent early-warning signals for predicting criti-cal transitions. However, computing Lyapunov expo-nents based on observed signal needs long enoughtime series, which should be as clean as possible.

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Fig. 6 a Bifurcationdiagram and b entropymeasure in ADD attentionmodel with respect tochanging A parameter(B = 5.821, w1 =1.487, w2 = 0.2223)

5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

A

Entro

py

a

b

Application of Lyapunov exponent for anticipating tip-ping points is illustrated by two biological models andone ecological model. The results show that Lyapunovexponents can be good predictors in different criticaltransitions.

Acknowledgements The authorswould like to thankDr.LeslieSamuel Smith for the help and edits.

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